(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

                    Mathematica-Compatible Notebook

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
Splitting blocks: A new vertex can be added to the center of a BBBB block \
without breaking continuity with neighboring blocks by adjusting vertex \
weights as follows. This generalizes to all other block types by replacing \
vectors along vector interpolated edges with the interpolated vector, and not \
adjusting the vertex weights for that edge.\
\>", "Text"],

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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            {"0", "0", "0", "0", "1", "1", "1", "1"},
            {"0", "0", "0", "0", "1", "1", "1", "1"},
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}, Open  ]],

Cell["\<\
The index of which weight is largest for all dyadic block dimensions between \
2 and 8 using the various modified blending weights. All assume the \
upper-left corner is the exterior corner.\
\>", "Text"],

Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Reflections", "Subsubsection"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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      MatrixForm\)], "Input"],

Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(u[0]\), \(u[1]\), \(u[2]\), \(u[3]\)},
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}, Open  ]],

Cell[BoxData[
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        w[3] \[Rule] i*j, u[0] \[Rule] 1 - j, u[1] \[Rule] i, 
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Cell[BoxData[
    \(w[0] := \((1 - i)\)*\((1 - j)\)\)], "Input"],

Cell[BoxData[
    \(w[1] := i*\((1 - j)\)\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
    \(w[3] := i*j\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
    \(u[3] := j\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(reflections[{m[0], m[1], m[3], m[2]}]\)], "Input"],

Cell[BoxData[
    \({{m[0], m[1], m[3], m[2]}, {m[1], m[3], m[2], m[0]}, {m[0], m[2], m[3], 
        m[1]}, {m[2], m[3], m[1], m[0]}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
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        exp_] := \({reflections[exp], \(D[#, i] &\) /@ 
              reflections[exp], \(D[#, j] &\) /@ 
              reflections[exp], \(D[#, i, j] &\) /@ reflections[exp]} /. 
          i -> 0\) /. j -> 0\)], "Input"],

Cell[BoxData[
    \(SetAttributes[deltas, HoldAll]\)], "Input"],

Cell[BoxData[
    \(testedges := \(\((Simplify[
                Expand[{# /. j -> 0, # /. i -> 1, # /. i \[Rule] 0, # /. 
                      j \[Rule] 1}]])\) &\)[Simplify[#]] &\)], "Input"],

Cell["VVVV", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(testedges[src[m[0]*w[0] + m[1]*w[1] + m[2]*w[2] + m[3]*w[3]]] // 
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Cell[BoxData[
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            m[ 0], 
            Times[ -1, i, 
              m[ 0]], 
            Times[ i, 
              m[ 1]]]], 
        src[ 
          Plus[ 
            m[ 1], 
            Times[ -1, j, 
              m[ 1]], 
            Times[ j, 
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        src[ 
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            Times[ -1, j, 
              m[ 0]], 
            Times[ j, 
              m[ 2]]]], 
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            m[ 2], 
            Times[ -1, i, 
              m[ 2]], 
            Times[ i, 
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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                m[3]\), \(m[0] - m[1] - m[2] + m[3]\), \(m[0] - m[1] - m[2] + 
                m[3]\)}
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}, Open  ]],

Cell["BVVV", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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          src[u[0]*m[0] + w[2]*m[2] + w[3]*m[3]]*u[2] + 
            src[u[0]*m[1] + w[2]*m[2] + w[3]*m[3]]*u[1]]] // 
      MatrixForm\)], "Input"],

Cell[BoxData[
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                  j\ m[3]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[1]] + 
                j\ src[m[3]]\), \(src[m[1] - j\ m[1] + j\ m[3]]\), \(src[
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            {\(src[m[0] - j\ m[0] + j\ m[2]]\), \(src[
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                j\ src[m[2]]\), \(src[m[0] - j\ m[0] + j\ m[2]]\)},
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                m[2] - i\ m[2] + 
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                i\ src[m[3]]\)}
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      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            {\(\(-m[0]\) + m[2]\), \(\(-m[0]\) + m[2]\), 
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      Function[ BoxForm`e$, 
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            {\(\(-m[1]\) + m[2]\), \(\(-m[0]\) + m[2]\), 
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      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["BVVB", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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          src[u[0]*m[0] + u[3]*m[2]]*u[2] + 
            src[u[0]*m[1] + u[3]*m[3]]*u[1]]] // MatrixForm\)], "Input"],

Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
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                i\ src[m[1]]\)},
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                j\ src[m[3]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[1]] + 
                j\ src[m[3]]\), \(src[m[1] - j\ m[1] + j\ m[3]]\)},
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                  j\ m[2]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[0]] + 
                j\ src[m[2]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[0]] + 
                j\ src[m[2]]\), \(src[m[0] - j\ m[0] + j\ m[2]]\)},
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                  i\ m[3]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[2]] + 
                i\ src[m[3]]\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
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            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
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            {\(\(-m[1]\) + m[3]\), "0", "0", \(\(-m[1]\) + m[3]\)},
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      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["BVBV", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(testedges[
        reflections[
          src[m[0]]*w[0] + src[u[0]*m[1] + u[3]*m[3]]*w[1] + 
            src[u[2]*m[2] + u[1]*m[3]]*w[2] + 
            src[w[1]*m[1] + w[2]*m[2] + \((w[0] + w[3])\)*m[3]]*w[3]]] // 
      MatrixForm\)], "Input"],

Cell[BoxData[
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            {\(\(-\((\(-1\) + i)\)\)\ src[m[0]] + 
                i\ src[m[1]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[0]] + 
                i\ src[m[1]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[0]] + 
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                j\ src[m[3]]\), \(src[m[1] - j\ m[1] + j\ m[3]]\), \(src[
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                  j\ m[2]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[0]] + 
                j\ src[m[2]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[0]] + 
                j\ src[m[2]]\)},
            {\(src[m[2] - i\ m[2] + i\ m[3]]\), \(src[
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                i\ src[m[3]]\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(m[1]\), \(m[2]\), \(m[2]\), \(m[1]\)},
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            {\(\(-m[1]\) + m[3]\), "0", "0", \(\(-m[1]\) + m[3]\)},
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            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(m[2]\), \(m[0]\), \(m[1]\), \(m[0]\)},
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            {"0", \(\(-m[0]\) + m[2]\), \(\(-m[1]\) + m[3]\), "0"},
            {"0", "0", "0", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(deltas[w[1]*m[1] + w[2]*m[2] + \((w[0] + w[3])\)*m[3]] // 
      MatrixForm\)], "Input"],

Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(m[3]\), \(m[0]\), \(m[3]\), \(m[0]\)},
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                m[3]\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["BBBV", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(testedges[
        reflections[
          src[m[0]]*w[0] + src[m[1]]*w[1] + 
            src[u[2]*m[2] + u[1]*m[3]]*u[3]]] // MatrixForm\)], "Input"],

Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
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                i\ src[m[1]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[0]] + 
                i\ src[m[1]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[0]] + 
                i\ src[m[1]]\), \(src[m[0] - i\ m[0] + i\ m[1]]\)},
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                j\ src[m[3]]\), \(\(-\((\(-1\) + j)\)\)\ src[m[1]] + 
                j\ src[m[3]]\), \(src[
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                j\ src[m[3]]\)},
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                j\ src[m[2]]\)},
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                  i\ m[3]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[2]] + 
                i\ src[m[3]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[2]] + 
                i\ src[m[3]]\), \(\(-\((\(-1\) + i)\)\)\ src[m[2]] + 
                i\ src[m[3]]\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(m[2]\), \(m[0]\), \(m[1]\), \(m[0]\)},
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            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
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}, Open  ]],

Cell["BBBB", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            {\(\(-\((\(-1\) + i)\)\)\ src[m[0]] + i\ src[m[1]]\)},
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            {\(\(-\((\(-1\) + j)\)\)\ src[m[0]] + j\ src[m[2]]\)},
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        Plus[ 
          Times[ -1, 
            Plus[ -1, i], 
            src[ 
              m[ 0]]], 
          Times[ i, 
            src[ 
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        Plus[ 
          Times[ -1, 
            Plus[ -1, j], 
            src[ 
              m[ 1]]], 
          Times[ j, 
            src[ 
              m[ 3]]]], 
        Plus[ 
          Times[ -1, 
            Plus[ -1, j], 
            src[ 
              m[ 0]]], 
          Times[ j, 
            src[ 
              m[ 2]]]], 
        Plus[ 
          Times[ -1, 
            Plus[ -1, i], 
            src[ 
              m[ 2]]], 
          Times[ i, 
            src[ 
              m[ 3]]]]}]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            {\(-1\), "1", \(-1\), "0"},
            {\(-1\), "0", \(-1\), "1"},
            {"1", \(-1\), "1", \(-1\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            {"1", "0", "0", "0"},
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            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
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            {"0", \(-1\), "1", \(-1\)},
            {"1", \(-1\), "0", \(-1\)},
            {\(-1\), "1", \(-1\), "1"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"0", "0", "0", "0"},
            {"0", "0", "0", "1"},
            {"0", "1", "0", "0"},
            {"1", \(-1\), "1", \(-1\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
Splitting blocks: A new vertex can be added to the center of a BBBB block \
without breaking continuity with neighboring blocks by adjusting vertex \
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